Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| isoclinism | isoclinic class of D81 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
| direct product | C22×D81 |
| unambiguous semidirect product | C81⋊C23 |
| unambiguous semidirect product | (C2×C81)⋊C22 |
| Fitting | C22×C81 ⋊ C2 |
| derived, Sylow | C81 ⋊ C23 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D81 |
| commutator, cosocle, hypocenter | C81 | C23 |
| Frattini | C27 | C22×D3 |
| Fitting | C22×C81 | C2 |
| socle | C22×C3 | D27 |
| LCS | G ▷ C81 | |
| UCS | C1 ◁ C22 | |
80 conjugacy classes of subgroups.
| transitive degree | μt(G) = 324 |
168 conjugacy classes — view the full character table.
| maximal subgroup of | (C2×D81)⋊C4, C2×C4×D81, C2×D324, … |
| maximal normal subgroup of | (C2×D81)⋊C4, C2×C4×D81, C2×D324, … |
| maximal quotient of | C2×Dic162, C2×C4×D81, C2×D324, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 43);